6. The breadth and thickness of a beam are 3 ft. 3 in. and 2 ft, 4in.; find by duodecimals the length of a piece which contains 14 cubic feet. Ans. 1 feet.

7. Find the size of a cube whose solidity is equal to that of a parallelopipedon whose dimensions are 1 ft. 8 in., 2 ft. 8 in., 4 ft. 1 in. Ans. 2 ft. 4 in.

8. The three contiguous edges of a parallelopipedon are 8, 10, 12; find the lengths of three lines drawn from one corner to the middle points of the opposite faces.

Ans. 11*180, 12°324, 13.601.

9. The length and breadth of a 56 lb. weight are 8 and 6 inches; find the height, allowing 2 lbs. for the handle which is sunk into the weight. Ans. 45 inches.

THE PRISM.

This solid is contained by rectangular planes, all parallel to the same straight line; the ends are equal rectilineal figures.

XXV.

To find the solidity of a prism. Multiply the area of the end by the length.

T. M.

5

Ex. Find the solidity of a pentagonal prism whose

length is 7 ft. 3 in.; the side of the pentagon being 3 ft. 6 in.

I. Find the solidity of a prism whose length is 20 ft. 6 in.; the end being an equilateral triangle whose side is 14 ft. 10 in. Ans. 1953 076 feet.

2.

Find the solidity of a triangular prism whose length is 12 ft. 9 in.; the sides of the end being 7 ft. 3 in., 8 ft. 4 in., 9 ft. 5 in.

Ans. 370 211 feet.

3. Find the solidity of an hexagonal prism whose length is 10 ft. 10 in.; the side of the hexagon being 5 ft. 5 in. Ans. 825 782 feet.

4. Find the solidity of a triangular prism whose length is 6 feet; the sides of the triangle being 2 ft. 1 in., 3 ft. 3 in., 4 ft. 8 in.

Ans. 17 feet.

THE CYLINDER.

This solid is formed by the revolution of a rectangle about one of its sides.

XXVI.

To find the curve surface of a cylinder.

Multiply the length by the diameter; then multiply

by 3.

Ex.

Find the curve surface of a cylinder whose length

is 10 ft. 3 in., and diameter 3 ft. 11 in.

To find the solidity of a cylinder.

Multiply the square of the diameter by the length; then multiply by 11 and divide by 14.

Ex. Find the solidity of a cylinder whose length is 4 ft. 8 in., and diameter 2 ft. 6 in.

CYLINDRICAL SHELL.

XXVIII.

To find the solidity of a cylindrical shell.

Multiply the sum of the inner and outer diameters by their difference, and this product by the length of the shell; then multiply by 11 and divide by 14.

Ex. Find the weight of a cast-iron pipe whose length is 9 feet, and the inner and outer diameters 6 and 8 inches. 8+6=14 8-6= 2

28

108

224

280

3024

3024

2 33264

7 16632

4 2376

594

Ans. 5 cwt. I qr. 6 lbs.